arXiv subjects

Giovanna Cerami

Publications and source records attributed to Giovanna Cerami.

4 recordsLinked to original sources

arXiv2018

Infinitely many positive standing waves for Schrödinger equations with competing coefficients

The paper deals with the equation $-Δu+a(x) u +b(x)u^q -u^p = 0$, $u \in H^1(\R^N)$, whith $N\ge 2$, $1 0$, $a(x)\to a_\infty$ and $b(x)\to 0$ as $|x|\to\infty$. When $a(x)\le a_\infty$ and $b(x) = 0$ only a finite number of positive solutions to the problem is reasonably expected. Here we prove that the presence of a nonzero term $b(x)u^q $ with $b(x)\geq 0, \ b(x)\neq 0,$ under suitable assumptions on the decay rates of $a$ and $b,$ allows to obtain infinitely many positive solutions.

math.AP↗

arXiv2018

Multiple positive bound states for critical Schrödinger-Poisson systems

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -Δu+V(x)u+K(x)ϕ(x)u=u^5 -Δϕ=K(x)u^2\qquad x\in\R^3 \end{array}\right.\quad\quad (SP) $$ We remark that $(SP)$ exhibits a "double" lack of compactness because of the unboundedness of $\R^3$ and the critical growth of the nonlinear term and that in our assumptions ground state solutions of $(SP)$ do not exist.

math.AP↗

arXiv2015

On Some Scalar Field Equations with Competing Coefficients

This paper deals with semilinear elliptic problems of the type \[ \left\{ \begin{array}{ll} -Δu+α(x)u= β(x)|u|^{p-1}u \quad \hbox{in }\mathbb{R}^N, u(x)>0\quad\hbox{in } \mathbb{R}^N, \qquad u \in H^1(\mathbb{R}^N), \end{array} \right. \] where $p$ is superlinear but subcritical and the coefficients $α$ and $β$ are positive functions such that $α(x) \to a_\infty > 0$ and $β(x)\to b_\infty > 0$, as $|x| \to \infty$. Aim of this work is to describe some phenomena that can occur when the coefficients are "competing".

math.AP↗

arXiv2013

Multiple Solutions for Scalar Field Equations with Potentials having "Subsidences"

In this paper the question of finding infinitely many solutions to the problem $-Δu+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that $a(x)$ satisfies a suitable "slow decay at infinity" condition and, moreover, that its graph has some "dips", we prove that the problem admits either infinitely many nodal solutions either infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.

math.AP↗